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May  2016, 15(3): 761-794. doi: 10.3934/cpaa.2016.15.761

## A weighted $L_p$-theory for second-order parabolic and elliptic partial differential systems on a half space

 1 Department of mathematics, Korea university, 1 anam-dong, sungbuk-gu, Seoul 136-701, South Korea 2 Department of Mathematics, Ajou University, 206 Worldcup-ro, Yeontong-gu, Suwon 443-749, South Korea

Received  March 2015 Revised  December 2015 Published  February 2015

In this article we consider parabolic systems and $L_p$ regularity of the solutions. With zero boundary condition the solutions experience bad regularity near the boundary. This article addresses a possible way of describing the regularity nature. Our space domain is a half space and we adapt an appropriate weight into our function spaces. In this weighted Sobolev space setting we develop a Fefferman-Stein theorem, a Hardy-Littlewood theorem and sharp function estimations. Using these, we prove uniqueness and existence results for second-order elliptic and parabolic partial differential systems in weighed Sobolev spaces.
Citation: Kyeong-Hun Kim, Kijung Lee. A weighted $L_p$-theory for second-order parabolic and elliptic partial differential systems on a half space. Communications on Pure & Applied Analysis, 2016, 15 (3) : 761-794. doi: 10.3934/cpaa.2016.15.761
##### References:
 [1] M. Bramanti and M. C. Ceruti, $W^{1,2}_p$ solvability for the Cauchy-Dirichlet problem for parabolic equations with VMO doeffieients,, \emph{Comm. Partial Differential Equations}, 18 (1993), 1735.  doi: 10.1080/03605309308820991.  Google Scholar [2] Sun-Sig Bun, Parabolic equations with BMO coefficients in Lipschitz domains,, \emph{J. Differential Equations}, 209 (2005), 229.  doi: 10.1016/j.jde.2004.08.018.  Google Scholar [3] F. Chiarenza, M. Frasca and P. Longo, $W^{2,p}$-solvability of the Dirichlet problem for nondivergence elliptic equations with VMO coefficients,, \emph{Trans. Amer. Math. Soc.}, 336 (1993), 841.  doi: 10.2307/2154379.  Google Scholar [4] P. Grisvard, Elliptic Problems in Nonsmooth Domains,, Monographs and Studies in Mathematics \textbf{24}, 24 (1985).   Google Scholar [5] R. Haller-Dintelmann, H. Heck and M. Hieber, $L^p-L^q$-estimates for parabolic systems in non-divergence form with VMO coefficients,, \emph{J. London Math. Soc.}, 74 (2006), 717.  doi: 10.1112/S0024610706023192.  Google Scholar [6] Doyoon Kim and N. V. Krylov, Parabolic equations with measurable coefficients,, \emph{Potential Anal.}, 26 (2007), 345.  doi: 10.1007/s11118-007-9042-8.  Google Scholar [7] N. V. Krylov, Some properties of traces for stochastic and deterministic parabolic weighted Sobolev spaces,, \emph{Journal of Functional Analysis}, 183 (2001), 1.  doi: 10.1006/jfan.2000.3728.  Google Scholar [8] N. V. Krylov, The heat equation in $L_p((0,T),L_p)$-spaces with weights,, \emph{SIAM J. Math. Anal.}, 32 (2001), 1117.  doi: 10.1137/S0036141000372039.  Google Scholar [9] N. V. Krylov, An analytic approach to SPDEs,, in \emph{Stochastic Partial Differential Equations: Six Perspectives}, 64 (1999), 185.  doi: 10.1090/surv/064/05.  Google Scholar [10] N. V. Krylov, Weighted Sobolev spaces and Laplace equations and the heat equations in a half space,, \emph{Comm. in PDEs}, 23 (1999), 1611.  doi: 10.1080/03605309908821478.  Google Scholar [11] N. V. Krylov, Some properties of weighted Sobolev spaces in $\bR^d_+$,, \emph{Ann. Scuola Norm. Sup. Pisa Cl. Sci.}, 28 (1999), 675.   Google Scholar [12] N. V. Krylov, A $W^n_2$-theory of the Dirichlet problem for SPDEs in general smooth domains,, \emph{Probab. Theory Relat. Fields}, 98 (1994), 389.  doi: 10.1007/BF01192260.  Google Scholar [13] N. V. Krylov, Lectures on Elliptic and Parabolic Equations in Sobolev Spaces,, American Mathematical Society, (2008).  doi: 10.1090/gsm/096.  Google Scholar [14] N. V. Krylov and S. V. Lototsky, A Sobolev space theory of SPDEs with constant coefficients on a half line,, \emph{SIAM J. Math. Anal.}, 30 (1999), 298.  doi: 10.1137/S0036141097326908.  Google Scholar [15] N. V. Krylov and S. V. Lototsky, A Sobolev space theory of SPDEs with constant coefficients in a half space,, \emph{SIAM J. on Math. Anal.}, 31 (1999), 19.  doi: 10.1137/S0036141098338843.  Google Scholar [16] Kijung Lee, On a deterministic linear partial differential system,, \emph{J. Math. Anal. Appl.}, 353 (2009), 24.  doi: 10.1016/j.jmaa.2008.11.059.  Google Scholar [17] J.-L. Lions and E. Magenes, Problèmes aux limites non homogénes et applications $1$,, Dunod, (1968).   Google Scholar [18] S. V. Lototsky, Sobolev spaces with weights in domains and boundary value problems for degenerate elliptic equations,, \emph{Methods and Applications of Analysis}, 1 (2000), 195.   Google Scholar [19] V. A. Solonnikov, Solvability of the classical initial-boundary-value problems for the heat-conduction equations in a dihedral angle,, \emph{Zapiski Nauchnykh Seminarov LOMI}, 138 (1984), 146.   Google Scholar [20] H. Triebel, Theory of Function Spaces,, Birkh\, (1983).  doi: 10.1007/978-3-0346-0416-1.  Google Scholar

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##### References:
 [1] M. Bramanti and M. C. Ceruti, $W^{1,2}_p$ solvability for the Cauchy-Dirichlet problem for parabolic equations with VMO doeffieients,, \emph{Comm. Partial Differential Equations}, 18 (1993), 1735.  doi: 10.1080/03605309308820991.  Google Scholar [2] Sun-Sig Bun, Parabolic equations with BMO coefficients in Lipschitz domains,, \emph{J. Differential Equations}, 209 (2005), 229.  doi: 10.1016/j.jde.2004.08.018.  Google Scholar [3] F. Chiarenza, M. Frasca and P. Longo, $W^{2,p}$-solvability of the Dirichlet problem for nondivergence elliptic equations with VMO coefficients,, \emph{Trans. Amer. Math. Soc.}, 336 (1993), 841.  doi: 10.2307/2154379.  Google Scholar [4] P. Grisvard, Elliptic Problems in Nonsmooth Domains,, Monographs and Studies in Mathematics \textbf{24}, 24 (1985).   Google Scholar [5] R. Haller-Dintelmann, H. Heck and M. Hieber, $L^p-L^q$-estimates for parabolic systems in non-divergence form with VMO coefficients,, \emph{J. London Math. Soc.}, 74 (2006), 717.  doi: 10.1112/S0024610706023192.  Google Scholar [6] Doyoon Kim and N. V. Krylov, Parabolic equations with measurable coefficients,, \emph{Potential Anal.}, 26 (2007), 345.  doi: 10.1007/s11118-007-9042-8.  Google Scholar [7] N. V. Krylov, Some properties of traces for stochastic and deterministic parabolic weighted Sobolev spaces,, \emph{Journal of Functional Analysis}, 183 (2001), 1.  doi: 10.1006/jfan.2000.3728.  Google Scholar [8] N. V. Krylov, The heat equation in $L_p((0,T),L_p)$-spaces with weights,, \emph{SIAM J. Math. Anal.}, 32 (2001), 1117.  doi: 10.1137/S0036141000372039.  Google Scholar [9] N. V. Krylov, An analytic approach to SPDEs,, in \emph{Stochastic Partial Differential Equations: Six Perspectives}, 64 (1999), 185.  doi: 10.1090/surv/064/05.  Google Scholar [10] N. V. Krylov, Weighted Sobolev spaces and Laplace equations and the heat equations in a half space,, \emph{Comm. in PDEs}, 23 (1999), 1611.  doi: 10.1080/03605309908821478.  Google Scholar [11] N. V. Krylov, Some properties of weighted Sobolev spaces in $\bR^d_+$,, \emph{Ann. Scuola Norm. Sup. Pisa Cl. Sci.}, 28 (1999), 675.   Google Scholar [12] N. V. Krylov, A $W^n_2$-theory of the Dirichlet problem for SPDEs in general smooth domains,, \emph{Probab. Theory Relat. Fields}, 98 (1994), 389.  doi: 10.1007/BF01192260.  Google Scholar [13] N. V. Krylov, Lectures on Elliptic and Parabolic Equations in Sobolev Spaces,, American Mathematical Society, (2008).  doi: 10.1090/gsm/096.  Google Scholar [14] N. V. Krylov and S. V. Lototsky, A Sobolev space theory of SPDEs with constant coefficients on a half line,, \emph{SIAM J. Math. Anal.}, 30 (1999), 298.  doi: 10.1137/S0036141097326908.  Google Scholar [15] N. V. Krylov and S. V. Lototsky, A Sobolev space theory of SPDEs with constant coefficients in a half space,, \emph{SIAM J. on Math. Anal.}, 31 (1999), 19.  doi: 10.1137/S0036141098338843.  Google Scholar [16] Kijung Lee, On a deterministic linear partial differential system,, \emph{J. Math. Anal. Appl.}, 353 (2009), 24.  doi: 10.1016/j.jmaa.2008.11.059.  Google Scholar [17] J.-L. Lions and E. Magenes, Problèmes aux limites non homogénes et applications $1$,, Dunod, (1968).   Google Scholar [18] S. V. Lototsky, Sobolev spaces with weights in domains and boundary value problems for degenerate elliptic equations,, \emph{Methods and Applications of Analysis}, 1 (2000), 195.   Google Scholar [19] V. A. Solonnikov, Solvability of the classical initial-boundary-value problems for the heat-conduction equations in a dihedral angle,, \emph{Zapiski Nauchnykh Seminarov LOMI}, 138 (1984), 146.   Google Scholar [20] H. Triebel, Theory of Function Spaces,, Birkh\, (1983).  doi: 10.1007/978-3-0346-0416-1.  Google Scholar
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